# A Scalable Nonparametric Continuous-Time Survival Model through Numerical Quadrature

arXiv: https://arxiv.org/abs/2605.16208 · HTML: https://mlsanity.com/paper/2605.16208/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 48 references that are in the map.
- First posted on arXiv 15 May 2026.

## Builds on (most cited first, 21 of 21)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Identifying the Best Machine Learning Algorithms for Brain Tumor Segmentation, Progression Assessment, and Overall Survival Prediction in the BRATS Challenge](https://mlsanity.com/paper/1811.02629/) (2018)
- [DeepSurv: Personalized Treatment Recommender System Using A Cox Proportional Hazards Deep Neural Network](https://mlsanity.com/paper/1606.00931/) (2016)
- [Time2Vec: Learning a Vector Representation of Time](https://mlsanity.com/paper/1907.05321/) (2019)
- [Time-to-Event Prediction with Neural Networks and Cox Regression](https://mlsanity.com/paper/1907.00825/) (2019)
- [Deep Learning for Survival Analysis: A Review](https://mlsanity.com/paper/2305.14961/) (2023)
- [A Scalable Discrete-Time Survival Model for Neural Networks](https://mlsanity.com/paper/1805.00917/) (2018)
- [Deep Survival Machines: Fully Parametric Survival Regression and Representation Learning for Censored Data with Competing Risks](https://mlsanity.com/paper/2003.01176/) (2020)
- [Effective Ways to Build and Evaluate Individual Survival Distributions](https://mlsanity.com/paper/1811.11347/) (2018)
- [Deep Survival Analysis](https://mlsanity.com/paper/1608.02158/) (2016)
- [Deep Neural Networks for Survival Analysis Based on a Multi-Task Framework](https://mlsanity.com/paper/1801.05512/) (2018)
- [Deep Cox Mixtures for Survival Regression](https://mlsanity.com/paper/2101.06536/) (2021)
- [Countdown Regression: Sharp and Calibrated Survival Predictions](https://mlsanity.com/paper/1806.08324/) (2018)
- [SODEN: A Scalable Continuous-Time Survival Model through Ordinary Differential Equation Networks](https://mlsanity.com/paper/2008.08637/) (2020)
- [Survival stacking: casting survival analysis as a classification problem](https://mlsanity.com/paper/2107.13480/) (2021)
- [Estimation of conditional mixture Weibull distribution with right-censored data using neural network for time-to-event analysis](https://mlsanity.com/paper/2002.09358/) (2020)
- [Survival Mixture Density Networks](https://mlsanity.com/paper/2208.10759/) (2022)
- [Survival Analysis via Ordinary Differential Equations](https://mlsanity.com/paper/2009.03449/) (2020)

## Abstract (arXiv)

Flexible continuous-time survival modeling is critical for capturing complex time-varying hazard dynamics in high-dimensional data; however, training such models remains challenging due to the intractable integral required for likelihood estimation. We introduce QSurv, a scalable deep learning framework that enables nonparametric continuous-time modeling without relying on time discretization or restrictive distributional assumptions. We propose a training objective based on Gauss-Legendre numerical quadrature, which approximates the cumulative hazard with high-order accuracy while facilitating efficient end-to-end training via standard backpropagation. Furthermore, to effectively capture non-stationary hazard dynamics in complex architectures, we introduce time-conditioned low-rank adaptation, a mechanism that conditions general neural backbones on time by dynamically modulating weights via low-rank updates. We provide theoretical analysis establishing approximation error bounds for cumulative-hazard evaluation. Comprehensive experiments across synthetic benchmarks, large-scale real-world tabular datasets, and high-dimensional medical imaging tasks demonstrate that QSurv achieves competitive predictive performance with advantages in instantaneous hazard function estimation, enabling more interpretable characterization of time-varying risk patterns.